The pole inflation from broken non-compact isometry in Weyl gravity

We propose the microscopic origin of the pole inflation from the scalar fields of broken non-compact isometry in Weyl gravity. We show that the $SO(1,N)$ isometry in the field space in combination with the Weyl symmetry relates the form of the non-minimal couplings to the one of the potential in the Jordan frame. In the presence of an explicit breaking of the $SO(1,N)$ symmetry in the coefficient of the potential, we realize the pole inflation near the pole of the inflaton kinetic term. Applying our results to the Higgs or PQ inflation models, we find that there is one parameter family of the solutions for the pole inflation, depending on the overall coefficient of the Weyl covariant derivatives for scalar fields. The same coefficient not only makes the predictions of the pole inflation varying, being compatible with the Planck data, but also determines the mass of the Weyl gauge field. We also show that the isocurvature perturbations of the axion can be suppressed sufficiently during the PQ pole inflation, and the massive Weyl gauge field produced during reheating serves as a dark matter candidate.